Concerning definitions used in this introduction we refer to section 2. Weisz [1]} proved the boundedness of the maximal operator of FejĂŠr means \(\sigma ^{\psi ,\ast }\)  with respect to bounded Vilenkin systems from the martingale Hardy space \(H_p(G_m)\)  to the space \(L_p(G_m),\)  for \(p>1/2\) . Simon [2]} gave a counterexample, which shows that boundedness does not hold for \(0<p<1/2.\)  The corresponding counterexample for \(p=1/2\)  is due to Goginava [3]}. Moreover, Weisz [4]} proved the following result:
